Control system, observer, and control method for a speed-sensorless induction motor drive

ABSTRACT

A feedback gain K is determined by using an estimation error e i  of a stator current, and an observed flux and an estimated speed are obtained and output based on the feedback gain K. An induction motor is controlled based on the observed flux and the estimated speed. In this way, a control system, an observer, and a control method, which realize a global stable control of an induction motor drive that does not comprise a speed sensor and/or a rotational position sensor, are implemented.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an induction motor drive without aspeed sensor and/or a rotational position sensor, and more particularly,to an observer for vector-controlling an induction motor drive.

2. Description of the Related Art

A typical vector control system for a direct field-oriented inductionmotor drive without a speed sensor and/or rotational position sensor isshown in FIG. 1.

In the system without a speed sensor, only a stator voltage 206 and astator current 207 are detected by sensors 108 and 109.

Vector control for an induction motor 100 in this figure is performedbased on the torque of the induction motor 100, which is independentlyapplied, and the magnetic flux fed by an inverter 101.

With the vector control in the system shown in this figure, a speedregulator 107 generates a torque current reference 202 under PI control(proportional action and integral action control) from a an estimatedspeed reference 200 being an instruction of the speed of the motor, andan estimated speed 211 from a flux and speed observer 110 as a feedback,and outputs the generated torque current reference 202 to a currentregulator 106. The current regulator 106 outputs a current that isregulated under the PI control from the torque current reference 202being an instruction to the torque and a flux current reference 201being an instruction to the magnetic flux. Then, a vector rotator 104transforms this current value into a relative value in a coordinatesystem (d-q coordinate system) that rotates in synchronization with asynthesized current vector, and applies the transformed value to theinverter 101 as a primary voltage command 205. The flux currentreference 201 applied to the current regulator 106 can be set to aconstant value over a wide operation range while the torque currentreference 202 is generated by a PI loop according to the estimated speed211.

The voltage and the current values applied from the inverter 101 to theinduction motor 100 are detected as the detected voltage 206 and thedetected current 207 by the sensors 108 and 109. After the detectedvoltage 206 and the detected current 207 are transformed into valuesrepresented by a two-phase coordinate system by 3-2 phase transformers102 and 103, they are input to the flux and speed observer 110 as spacevector values v_(s) 208 and i_(s) 209.

The flux and speed observer 110 obtains an observed rotor flux 210 fromthe stator voltage v_(s) 208 and the stator current i_(s) 209, outputsthe obtained flux 210 to vector rotators 104 and 105, estimates a rotorspeed, and outputs the estimated speed 211 to the speed regulator 107.

The vector rotator 104 vector-rotates a flux command 203 and a torquecommand 204 in the orientation of the rotor flux based on the observedrotor flux 210, and outputs the vector-rotated commands to the inverter101 as the primary voltage instruction 205.

Additionally, the vector i_(s) 209 is vector-rotated by the vectorrotator 105 in the orientation of the rotor flux based on the observedflux 210 from the flux and speed observer 110 in order to obtain a fluxcurrent 212 and a torque current 213, which are used as feedback signalsby the current regulator 106.

An MRAS (Model Reference Adaptive System) based on a flux and speedobserver was initially proposed by Ref. 1.

Ref. 1: H. Kubota et al. “DSP-based speed adaptive flux observer ofinduction motor”, IEEE Trans. Industry Applicat., vol. 2, no. 2 pp.343-348, 1993.

According to Ref. 1, a stator current and a rotor flux are used as anindependent set of variables in order to explain an induction motor.Accordingly, an equation for an induction motor, which is demonstratedby Ref. 1, can be rewritten to an equation using a stator flux and arotor flux as state variables. Since the process of this rewrite is astandard linear transformation, it is omitted here.

A classical representation of an induction machine in a stator orientedreference coordinate system (α-β) using a state space notation is asfollows. $\begin{matrix}\left\{ \begin{matrix}{{\frac{}{t}\left( \frac{\varphi_{s}}{\varphi_{r}} \right)} = {{{\begin{pmatrix}{{- R_{s}}L_{sg}I} & {R_{s}L_{m\quad g}I} \\{R_{r}L_{m\quad g}I} & {{{- R_{r}}L_{rg}I} + {\omega_{r}J}}\end{pmatrix} \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} + {\begin{pmatrix}I \\0\end{pmatrix} \cdot v_{s}}} = {{Ax} + {Bu}}}} \\{i_{s} = {{\left( {{L_{sg}I} - {L_{m\quad g}I}} \right) \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} = {Cx}}}\end{matrix} \right. & (1)\end{matrix}$

where:

φ_(s)=[φ_(sα) φ_(sβ)]^(T), φ_(r)=[φ_(rα) φ_(rβ)]^(T) , i _(s) =[i _(sα)i _(sβ)]^(T) , v _(s) =[v _(sα) v _(sβ)]^(T)

are space vectors associated with a stator flux, a rotor flux, a statorcurrent, and a stator voltage respectively.

Other symbols are as follows.$L_{sg} = {\frac{1}{\sigma \cdot L_{s}} = \frac{L_{r}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}$$L_{rg} = {\frac{1}{\sigma \cdot L_{r}} = \frac{L_{s}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}$$L_{m\quad g} = {\frac{1 - \sigma}{\sigma \cdot L_{m}} = \frac{L_{m}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}$${I = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}};\quad {J = \begin{bmatrix}0 & {- 1} \\1 & 0\end{bmatrix}};\quad {0 = \begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}};$

R_(s), R_(r): Stator and rotor resistance;

L_(s), L_(r), L_(m): Stator, rotor, and mutual inductance;

σ=1−L_(m) ²/(L_(s)·L_(r)): Total leakage coefficient;

ω_(r): Angular rotor speed.

Furthermore, according to Ref. 1, observed flux values are representedas follows. Note that observation and an observed value referred to inthis specification represent observation and an observed value in moderncontrol theory, and indicate the estimation of a state variable valuefrom an output and its estimated value. In the following equation,observed values are marked with “{circumflex over ( )}”. $\begin{matrix}\left\{ \begin{matrix}{{\frac{}{t}\begin{pmatrix}{\hat{\varphi}}_{s} \\{\hat{\varphi}}_{r}\end{pmatrix}} = {{\begin{pmatrix}{{- R_{s}}L_{sg}I} & {R_{s}L_{m\quad g}I} \\{R_{r}L_{m\quad g}I} & {{{- R_{r}}L_{rg}I} + {{\hat{\omega}}_{r}J}}\end{pmatrix} \cdot \begin{pmatrix}{\hat{\varphi}}_{s} \\{\hat{\varphi}}_{r}\end{pmatrix}} + {\begin{pmatrix}I \\0\end{pmatrix} \cdot v_{s}} +}} \\{{\begin{pmatrix}{{k_{1}I} + {k_{2}J}} \\{{k_{3}I} + {k_{4}J}}\end{pmatrix} \cdot \left( {{\hat{i}}_{s} - i_{s}} \right)} = {{\hat{A}\quad \hat{x}} + {B\quad u} + {K\quad e_{i}}}} \\{{\hat{i}}_{s} = {{\left( {{L_{sg}I} - {L_{m\quad g}I}} \right) \cdot \begin{pmatrix}{\hat{\varphi}}_{s} \\{\hat{\varphi}}_{r}\end{pmatrix}} = {C\hat{x}}}}\end{matrix} \right. & (2)\end{matrix}$

An output feedback gain K in the equation (2) is used to modify thedynamic characteristics of an estimation error and its determination.

The speed is evaluated with the following equation. $\begin{matrix}{{\frac{}{t}{\hat{\omega}}_{r}} = {{{k_{\omega} \cdot \left( {{\hat{i}}_{s} - i_{s}} \right)} \times {\hat{\varphi}}_{r}} = {k_{\omega} \cdot {\hat{\varphi}}_{r}^{T} \cdot J \cdot e_{i}}}} & (3)\end{matrix}$

where k_(ω) is an arbitrary gain.

According to Ref. 1, the feedback gain K in the observer equation (2) isused along with a constant of proportionality k in order to obtain foureigenvalues λ_(obs) of the flux observer represented by the equation(2), which are proportional to an eigenvalue λ_(mot) of thecorresponding motor represented by the equation (1).

λ_(obs) =k·λ _(mot)  (4)

The equation (4) is proved in the following document.

Ref. 2: Y. Kinpara and M. Koyama, “Speed Sensorless Vector ControlMethod of Induction Motor Including A Low Speed Region,” The Journal “D”of the Institute of Electrical Engineers of Japan, vol. 120-D, no. 2,pp. 223-229, 2000.

With a selection method for a feedback gain K, which is proposed by Ref.2, several unstable operation conditions are imposed on the inductionmotor. Especially, when a stator frequency approaches “0”, an observerdoes not converge, leading to inability of the operations of the motordrive.

An unstable region on a torque-speed plane of the induction motor drivedepends on the value of the constant of proportionality k in theequation (4). This unstable area converges to a single linecorresponding to the primary frequency that is exactly “0” when theconstant k converges to “0”. Accordingly, the dynamic characteristicsresultant from the flux observer become unacceptably slow for a verysmall value of k. Therefore, this selection method for the feedback gainK is not a good solution.

Ref. 2 proposes a method based on a Riccati equation as a selectionmethod for the output feedback gain K in the equation (2), whichstabilizes the drive.

With this method, if G, Q, and R are defined as follows${G = \begin{bmatrix}0 & 0 \\0 & J\end{bmatrix}};\quad {Q = \begin{bmatrix}I & 0 \\0 & I\end{bmatrix}};$

the output feedback gain K is obtained with the following equation.

K=PC ^(T) R ⁻¹

where P is a sole positive definite solution that satisfies thefollowing equation.

PA ^(T) +AP−PC ^(T) R ⁻¹ CP+GQG ^(T)=0

With the method using the Riccati equation, the stability of the fluxand speed observer is improved, but one arbitrary parameter (ε₁) that isnot directly related to the stability of a global operation must beselected to obtain the output feedback gain. If this parameter isunsuitably selected, the observer can possibly be made unstable or anunacceptably large delay can possibly be caused. In either case, aresultant operation cannot run at a very low primary frequency.

SUMMARY OF THE INVENTION

The present invention was developed to overcome the above describedproblems, and aims at providing a control system, an observer, and acontrol method for an induction motor drive without a speed sensor or aposition sensor, the operations of which are stable for global operationfrequencies.

The present invention assumes a device or a method performing vectorcontrol for an induction motor that does not comprise at least either aspeed sensor or a position sensor.

The control system according to the present invention comprises anobserver unit and a control unit.

The observer unit determines a feedback gain K by using an estimationerror of a stator current, and obtains and outputs at least either of anobserved magnetic flux and an estimated speed based on the feedback gainK.

The control unit controls the induction motor based on the output of theobserver unit.

With this system, only the restriction on the determination of thefeedback gain K is, for example, an equation${\lim\limits_{\omega\rightarrow 0}{\angle \quad {G_{l}({j\omega})}}} = \infty$

That is, the feedback gain K which satisfies a condition based on adifferent factor can be determined with almost no restrictions.

Furthermore, the observer unit can be configured to determine thefeedback gain K the magnitude of which is within a predetermined range,if the rotation speed of the induction motor is equal to or higher thana preset speed.

With this configuration, a stable operation can be realized even at anoperating frequency in the vicinity of “0”.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a typical system of a direct field-oriented induction motordrive without a speed sensor and/or rotational position sensor;

FIG. 2 exemplifies an output error block according to a preferredembodiment of the present invention;

FIG. 3 shows a system of a direct field-oriented induction motor drivewithout a speed sensor or a rotational position sensor according to thepreferred embodiment;

FIG. 4 exemplifies the configuration of a flux and speed observer; and

FIG. 5 exemplifies the simplest configuration of a stabilizing gaincalculator.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Preferred embodiments according to the present invention will beexplained below with reference to the drawings.

The present invention adopts several results established so far innon-linear control theory. In consequence, a feedback gain K thatensures the stability of a flux and speed observer under every possibleoperating condition can be obtained without much degrading the dynamiccharacteristics of the observer.

Since the feedback gain K is obtained as a very simple function ofalmost no motor parameters and operating speed, its realization andimplementation can be made with significant ease. Furthermore, unlikethe methods proposed so far, a feedback gain the value of which does notbecome infinite can be derived even if the primary frequency approaches“0”, according to the present invention (singular condition).

If an error of an angular rotor speed is set as Δω_(r)={circumflex over(ω)}_(r)−ω_(r), an estimation error e_(i)=î_(s)−i_(s) of an estimatedstator output current becomes as follows from the equations (1) and (2):

e _(i) =Ce=C(sI ₄ −A−KC)⁻¹(_(−I) ⁰)(−({circumflex over(ω)}_(r)−ω_(r))J{circumflex over (φ)} _(r))=G _(l)(s)·(−Δω_(r)J{circumflex over (φ)} _(r))  (5)

where s is a Laplace operator.

FIG. 2 exemplifies an output error block satisfying the equation (5).

As exemplified in FIG. 2, a stator current estimation error system canbe recognized as an interconnection of a non-linear feedback transferfunction and a linear transfer function.

If the following two conditions are satisfied in this figure, the fluxand speed observer is stabilized.

Condition 1: A non-linear feedback 20 shown in FIG. 2 satisfies thePopov's inequality that is well known as a stability determinationmethod for a non-linear control system. Namely, a constant γ which isnot dependent on a time t exists, and the following inequality issatisfied for every t₁>t₀. $\begin{matrix}{{\int_{0}^{t_{1}}{{{u_{nl}^{T}(\tau)} \cdot {y_{nl}(\tau)}}{\tau}}} \geq {- \gamma^{2}}} & (6)\end{matrix}$

Condition 2: A linear transfer function G_(l)(s) of a linearsteady-state block 10 shown in FIG. 2 is stable, and a phase anglebetween an input and an output falls within a range of ±π/2.

Since mechanical constituent elements have relatively slow dynamiccharacteristics as for Condition 1, it becomes easy to implementu_(nl)=e_(i), y_(nl)={circumflex over (ω)}_(r)J{circumflex over (φ)}_(r)in FIG. 2, if an actual speed is assumed to be nearly constant.

The following equation can be derived from the above provided equations(3) and (6). $\begin{matrix}{{u_{nl}^{T}y_{nl}} = {\frac{k_{\omega}}{s} \cdot \left( {e_{i}^{T}J\quad {\hat{\varphi}}_{r}} \right)^{2}}} & (7)\end{matrix}$

It is evident that the equation (7) satisfies Popov's inequality (6),since the equation (7) includes the feedback gain K. Hence, FIG. 2satisfies Condition 1.

Next, the linear transfer function G_(l)(s) in the equation (5) isconsidered as for Condition 2.

Suppose that there is no feedback gain K (K=0) In this case, althoughthe transfer function G_(l)(s) becomes stable, the phase displacementbetween an input and an output is outside the stable range of +π/2 forsufficiently low frequencies, which makes the observer unstable.

Accordingly, for the stability of the system shown in FIG. 2, it isnecessary to determine a gain matrix K such that the Popov's stabilitycondition is satisfied, and the whole of a dynamic matrix (A+KC) of theflux observer itself remains stable.

The transfer function G_(l)(s) in the equation (5) can be verified tochange its sign based on a primary frequency. This reflects on the phaseas follows. $\begin{matrix}{{{\lim\limits_{\omega\rightarrow{0 +}}\quad {\angle \quad {{\overset{\_}{G}}_{l}\left( {j\quad \omega} \right)}}}} = {\left. \alpha\Rightarrow{{\lim\limits_{\omega\rightarrow{0 -}}\quad {\angle \quad {{\overset{\_}{G}}_{l}\left( {j\quad \omega} \right)}}}} \right. = {\pi - \alpha}}} & (8)\end{matrix}$

This equation means that the only way to achieve the stability when theprimary frequency is a small positive or negative value is to ensure thefollowing equation. $\begin{matrix}{{{\lim\limits_{\omega\rightarrow 0}\quad {\angle \quad {{\overset{\_}{G}}_{l}\left( {j\quad \omega} \right)}}}} = {\pi/2}} & (9)\end{matrix}$

This equation (9) is satisfied whenever the feedback gain K thatsatisfies the following equation (10) is suitably selected.$\begin{matrix}{{\lim\limits_{\omega\rightarrow 0}\quad {\angle \quad {G_{l}\left( {j\quad \omega} \right)}}} = \infty} & (10)\end{matrix}$

As far as the equation (10) is satisfied, any feedback gain K in theform represented by the equation (2) is acceptable, and the flux andspeed observer is globally stabilized.

The feedback gain K can be defined as follows from the equation (2).$\begin{matrix}{K = \begin{pmatrix}{{k_{1}I} + {k_{2}J}} \\{{k_{3}I} + {k_{4}J}}\end{pmatrix}} & (11)\end{matrix}$

There are four parameters (k₁, . . . k₄), which can be arbitrarily set,for selecting the feedback gain K, and the only restriction on thisselection is the equation (10). Therefore, many options can be proposedfor a global stabilization problem, and a suitable one can be selectedfrom among the options, according to other conditions.

This selection method for the feedback gain K can be simplified, forexample, by selecting the parameters as follows.

k ₁ =k ₄=0  (12)

When it is verified that the parameter k₃ of the feedback gain K doesnot affect the restriction of the equation (10), only the gain parameterk₂ is proved to affect the restriction of the equation (10). Therefore,the feedback gain K that always satisfies the equation (10) can bederived.

Considering this fact, the parameter k₂ results in the followingequation (13). $\begin{matrix}{k_{2} = {k_{2C} = {{- L_{r}} \cdot \frac{R_{s}}{R_{r}} \cdot \omega_{r}}}} & (13)\end{matrix}$

From the above discussion, the following simple equation for derivingthe feedback gain K can be obtained. $\begin{matrix}{K = \begin{pmatrix}{k_{2C}J} \\0\end{pmatrix}} & (14)\end{matrix}$

The feedback gain K that satisfies the equation (14) stabilizes theobserver at any speed or at any primary frequency in which the flux andspeed observer has one of its poles at the origin (stability limit)except for the singular condition that the primary frequency, is exactly“0”.

It is proved from the equation (13) that the parameter k_(2c) of thefeedback gain K linearly becomes large with an increase in the operatingspeed. However, the operating frequency cannot be made very much low fora sufficiently high speed, due to the limitation of the slip of theinduction machine.

To address this problem, an upper limit is set for the value of thefeedback gain K, which is obtained from the equation (14), and the valueis clipped to fall within a particular range if it exceeds the range.For example, an operating speed approximately twice (or three times) therated slip ω_(s,rat) of the motor in use, such as a nominal value, etc.,is defined to be a maximum value, to which clipping is made, for anangular rotor speed ω_(r), so that the upper limit can be set for thefeedback gain K. In this way, no adverse effect is exerted on thestability of the drive at an operating frequency in the vicinity of “0”.

Which ever value the parameter k₃ of the feedback gain K takes, theequation (10) is satisfied. Therefore, the selection of the parameter k₃can be used to improve the dynamic characteristics or the stabilitymargin of the flux and speed observer represented by the equation (2).

Any standard technique may be available as the selection of theparameter k₃. For instance, the parameter k₃ can be obtained bylinearizing a system that is configured by the motor represented by theequation (1) and the observer represented by the equation (2) in thevicinity of an equilibrium operating point, or by using many tools thatcan be obtained from the established linear control theory.

Whichever value is selected for the parameter k₃, the observer remainsstable globally as far as the other parameters of the gain K, which areobtained from the equations (12) and (13), are not changed.

FIG. 3 shows the control system for a direct field-oriented inductionmotor drive without a speed sensor and/or a rotational position sensor,according to this preferred embodiment.

Comparing with the system configuration shown in FIG. 1, a flux andspeed observer 300 is arranged in the configuration shown in FIG. 3 as areplacement of the flux and speed observer 100. In this figure, the sameconstituent elements as those of the conventional system shown in FIG. 1are denoted with the same reference numerals, and their operations arefundamentally the same as those explained with reference to FIG. 1.

The globally stable flux and speed observer 300 according to thispreferred embodiment is used in a standard direct vector controlmechanism. The flux and speed observer 300 obtains an observed rotorflux 301, which is used by vector rotators 104 and 105 as a fieldorientation, with the use of a voltage vector v_(s) 208 and a currentvector i_(s) 209 that are transformed from a detected voltage 206 and adetected current 207, which are measured by the sensors 108 and 109,into a two-phase coordinate system by 3-2 phase transformers 102 and103, and outputs the obtained flux to the vector rotators 104 and 105.The flux and speed observer 300 is configured as a single unit in theconfiguration shown in FIG. 3. However, the flux and speed observer 300may be arranged separately as a flux observer and a speed observer.

Additionally, the flux and speed observer 300 obtains and outputs anestimated rotor speed 311 from the voltage vector v_(s) 208 and thecurrent vector i_(s) 209. The estimated rotor speed 311 is used as anexternal speed control loop 211 for a speed regulator 107.

FIG. 4 exemplifies the configuration of the flux and speed observer 300shown in FIG. 3.

In the configuration shown in FIG. 4, the observed rotor flux 310 andthe estimated rotor speed 311 are obtained from the voltage vector v_(s)208 and the current vector i_(s) 209 based on the above providedequations (2) and (3), and output.

In the flux and speed observer 300, an arithmetic operation unit 411calculates the difference between an input measured current vector i_(s)209 and an observed value 327 that the flux and speed observer 300itself calculates, and outputs the calculated difference to multipliers406 and 410 as an estimation error e_(i) 325 of the stator current.

The multiplier 406 multiplies the estimation error e_(i) 325 and thefeedback gain K obtained by a stabilizing gain calculator 400 to bedescribed later, and outputs the result of the multiplication to anadder 404.

Additionally, in the flux and speed observer 300, a motor estimator 401obtains the values of the first and the second terms in the equation (2)from the input voltage vector v_(s) 208 and the estimated rotor speed311 that the flux and speed observer 300 itself calculates. Then, theadder 404 outputs the value obtained by adding the value of the thirdterm Ke_(i) input from the multiplier 406 to the sum of the first andthe second terms in the equation (2) from the motor estimator 401. Thisvalue is integrated by an integrator 403.

The output of the integrator 403 is input to multipliers 402 and 405,and the motor estimator 401. The motor estimator 401 uses the output ofthe integrator 403 to obtain the first term of the equation (2). Themultiplier 402 multiplies the output of the integrator 403 and a fixedvalue matrix [L_(sg)I−L_(mg)I], which is dependent on thecharacteristics of the motor, so as to obtain the observed value 327 ofthe stator current, which is used by the arithmetic operation unit 411to obtain the estimation error e_(i) 325 of the stator current. Themultiplier 405 multiplies the output of the integrator 403 and a matrix[0(0 matrix) I] so as to generate an output value 310.

The multiplier 410 multiplies the estimation error e_(i) 325 of thestator current, and the result of the multiplication, which is outputfrom the multiplier 409, of the observed rotor flux 310 transposed by atransposer 412 and a matrix J 323, and outputs the result of themultiplication to a multiplier 408. The multiplier 408 multiplies thisvalue and an arbitrary positive gain k_(ω) in the equation (3). Anintegrator 407 integrates this result to obtain the estimated rotorspeed 311.

The stabilizing gain calculator 400 obtains the feedback gain K 326 forstabilization, which satisfies the equation (10), from the estimatedrotor speed 311 that the flux and speed observer 300 itself calculatesand fixed parameters indicating the characteristics of the motor.

FIG. 5 exemplifies the simplest configuration of the stabilizing gaincalculator 400 shown in FIG. 4.

The stabilizing gain calculator 400 having the configuration shown inFIG. 5 sets the parameters k₁, k₃, and k₄ to 0 among the four parametersk₁ to k₄ of the feedback gain K 326, and obtains only the parameter k₂from the estimated rotor speed 311.

Calculation of the parameter k₂ is made based on the equation (13). Ifan input absolute value of the estimated rotor speed 311 is equal to orlarger than a limiter value ω_(lim), a limiter 501 clips the estimatedrotor speed 311 to the limiter value ω_(lim) as represented by a graphof FIG. 5 to make the absolute value fall within the limiter valuerange, and outputs the clipped estimated rotor speed 311, inconsideration of the case where the equation (10) is not satisfied dueto the estimated rotor speed 311 used for the calculation, which is toohigh or low.

Then, a multiplier 502 multiplies this output value 504 and a fixedvalue −L_(r)·R_(s)/R_(r) 505, which is dependent on the standardparameters of the induction motor 100, to obtain the parameter k₂. Anarithmetic operation unit 500 calculates the stabilization feedback gainK 326 from the parameter k₂ and the values 503 (all the values are 0) ofk₁, k₃, and k₄, and outputs the feedback gain K 326.

As described above in detail, stable operations can be realized forglobal operating frequencies according to the present invention.

Additionally, since the procedures proposed to evaluate the feedbackgain of an MRAS-based flux and speed observer are used, it becomepossible to overcome problems that are associated with a regenerativeoperation at a low speed even if a primary frequency approaches 0, whichmakes a flux and speed observer causes an error with a conventionalmethod. As a result, global stable operations of the drive can beachieved.

Furthermore, a feedback gain the value of which does not become infinitecan be derived even if the primary frequency approaches “0”.

Still further, a stabilization gain can be obtained by solving a simplealgebraic equation.

Still further, a feedback gain K is obtained as a very simple functionof almost no motor parameters and operating speed. Therefore, thefeedback gain K that ensures the stability of a flux and speed observerunder every possible operating condition can be obtained without muchdegrading the dynamic characteristics of the observer. Accordingly, itis very easy to industrially realize and implement the presentinvention.

What is claimed is:
 1. A control system vector-controlling an inductionmotor that does not comprise at least either a speed sensor or aposition sensor, and is represented by a following equation,$\left\{ \begin{matrix}{{\frac{}{t}\left( \frac{\varphi_{s}}{\varphi_{r}} \right)} = {{{\begin{pmatrix}{{- R_{s}}L_{sg}I} & {R_{s}L_{m\quad g}I} \\{R_{r}L_{m\quad g}I} & {{{- R_{r}}L_{rg}I} + {\omega_{r}J}}\end{pmatrix} \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} + {\begin{pmatrix}I \\0\end{pmatrix} \cdot v_{s}}} = {{Ax} + {Bu}}}} \\{i_{s} = {{\left( {{L_{sg}I} - {L_{m\quad g}I}} \right) \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} = {Cx}}}\end{matrix} \right.$

where φ_(s)=[φ_(sα) φ_(sβ)]^(T): space vectors associated with a statorflux; φ_(r)=[φ_(rα) φ_(rβ)]^(T): space vectors associated with a rotorflux; i_(s)=[i_(sα) i_(sβ)]^(T): space vectors associated with a statorcurrent; i_(r)=[i_(rα) i_(rβ)]^(T): space vectors associated with astator voltage;${L_{sg} = {\frac{1}{\sigma \cdot L_{s}} = \frac{L_{r}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{rg} = {\frac{1}{\sigma \cdot L_{r}} = \frac{L_{s}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{m\quad g} = {\frac{1}{\sigma \cdot L_{m}} = \frac{L_{m}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${I = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}};\quad {J = \begin{bmatrix}0 & {- 1} \\1 & 0\end{bmatrix}};\quad {0 = \begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}};$

R_(s), R_(r): Stator and rotor resistance; L_(s), L_(r), L_(m): Stator,rotor, and mutual inductance; σ=1−L_(m) ²/(L_(s)·L_(r)): Total leakagecoefficient; ω_(r): Angular rotor speed said control system comprising:an observer unit determining a feedback gain K which stabilizes anequation of estimation error of an estimated stator current e_(i),$\begin{matrix}{e_{i} = \quad {{Ce} = {{C\left( {{s\quad I_{4}} - A - {K\quad C}} \right)}^{- 1}\begin{pmatrix}0 \\{- I}\end{pmatrix}\quad \left( {{- \left( {{\hat{\omega}}_{r} - \omega_{r}} \right)}J\quad {\hat{\varphi}}_{r}} \right)}}} \\{= \quad {{G_{1}(s)} \cdot \left( {{- \Delta}\quad \omega_{r}J\quad \varphi_{r}} \right)}}\end{matrix}$

 where values below the symbol {circumflex over ( )} are observedvalues; C=[L_(sg)I−L_(mg)I]; ${A = \begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}};$

A₁₁=−R_(s)L_(sg)I; A₁₂=R_(s)L_(mg)I; A₂₁=R_(r)L_(mg)I;A₂₂=−R_(r)L_(rg)I+ω_(r)J; e_(i): estimated stator current; s: Laplaceoperator; G_(l)(s): linear transfer function; and obtaining andoutputting at least either of an observed flux and an estimated speedbased on the feedback gain K; and a control unit controlling theinduction motor based on an output of said observer unit.
 2. The controlsystem according to claim 1, wherein said observer unit determines avalue that satisfies${\lim\limits_{\omega\rightarrow 0}{\angle \quad {G_{l}({j\omega})}}} = \infty$

where ω is primary frequency, and ∠ is a phase angle as a feedback gainK for a transfer function G_(l) being a linear portion of a system ofthe estimation error of the stator current.
 3. The control systemaccording to claim 1, wherein said observer unit determines the feedbackgain K based on a following equation$k_{2C} = {{- L_{r}} \cdot \frac{R_{s}}{R_{r}} \cdot \omega_{r}}$$K = \begin{pmatrix}{k_{2C}J} \\0\end{pmatrix}$

where L_(r) is a rotor inductance, R_(s) is a stator resistance, R_(r)is a rotor resistance, ω_(r) is an angular rotor speed, and k_(2c) is aparameter of the feedback gain K and ${J = \begin{bmatrix}0 & {- 1} \\1 & 0\end{bmatrix}};\quad {0 = {\begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}.}}$


4. The control system according to claim 1, wherein said observer unitdetermines the feedback gain K a magnitude of which is within apredetermined range if a rotation speed of the induction motor is equalto or higher than a preset value.
 5. The control system according toclaim 4, wherein said observer unit determines the feedback gain K byusing a value of an angular rotor speed ω_(r), which is restricted towithin a predetermined range.
 6. The control system according to claim1, wherein said control unit comprises a vector rotating unit performingvector rotation based on the output of said observer unit, and a currentregulating unit outputting a current command to the induction motorbased on the output of said observer unit.
 7. A control systemvector-controlling an induction motor that does not comprise at leasteither a speed sensor or a position sensor, and is represented by afollowing equation, $\left\{ \begin{matrix}{{\frac{}{t}\left( \frac{\varphi_{s}}{\varphi_{r}} \right)} = {{{\begin{pmatrix}{{- R_{s}}L_{sg}I} & {R_{s}L_{m\quad g}I} \\{R_{r}L_{m\quad g}I} & {{{- R_{r}}L_{rg}I} + {\omega_{r}J}}\end{pmatrix} \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} + {\begin{pmatrix}I \\0\end{pmatrix} \cdot v_{s}}} = {{Ax} + {Bu}}}} \\{i_{s} = {{\left( {{L_{sg}I} - {L_{m\quad g}I}} \right) \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} = {Cx}}}\end{matrix} \right.$

where φ_(s)=[φ_(sα) φ_(sβ)]^(T): space vectors associated with a statorflux; φ_(r)=[φ_(rα) φ_(rβ)]^(T): space vectors associated with a rotorflux; i_(s)=[i_(sα) i_(sβ)]^(T): space vectors associated with a statorcurrent; i_(r)=[i_(rα) i_(rβ)]^(T): space vectors associated with astator voltage;${L_{sg} = {\frac{1}{\sigma \cdot L_{s}} = \frac{L_{r}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{rg} = {\frac{1}{\sigma \cdot L_{r}} = \frac{L_{s}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{m\quad g} = {\frac{1}{\sigma \cdot L_{m}} = \frac{L_{m}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${I = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}};\quad {J = \begin{bmatrix}0 & {- 1} \\1 & 0\end{bmatrix}};\quad {0 = \begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}};$

R_(s), R_(r): Stator and rotor resistance; L_(s), L_(r), L_(m): Stator,rotor, and mutual inductance; σ=1−L_(m) ²/(L_(s)·L_(r)): Total leakagecoefficient; ω_(r): Angular rotor speed, said control system comprising:observer means for determining a feedback gain K which stabilizes anequation of estimation error of an estimated stator current e_(i),$\begin{matrix}{e_{i} = \quad {{Ce} = {{C\left( {{s\quad I_{4}} - A - {K\quad C}} \right)}^{- 1}\begin{pmatrix}0 \\{- I}\end{pmatrix}\quad \left( {{- \left( {{\hat{\omega}}_{r} - \omega_{r}} \right)}J\quad {\hat{\varphi}}_{r}} \right)}}} \\{= \quad {{G_{1}(s)} \cdot \left( {{- \Delta}\quad \omega_{r}J\quad \varphi_{r}} \right)}}\end{matrix}$

 where values below the symbol {circumflex over ( )} are observedvalues; C=[L_(sg)I−L_(mg)I]; ${A = \begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}};$

A₁₁=−R_(s)L_(sg)I; A₁₂=R_(s)L_(mg)I; A₂₁=R_(r)L_(mg)I;A₂₂=−R_(r)L_(rg)I+ω_(r)J; e_(i): stator current; s: Laplace operator;G_(l)(s): linear transfer function; and for obtaining and outputting atleast either of an observed flux and an estimated speed based on thefeedback gain K; and control means for controlling the induction motorbased on an output of said observer means.
 8. An observer used tovector-control an induction motor that does not comprise at least eithera speed sensor or a position sensor, and is represented by a followingequation, $\left\{ \begin{matrix}{{\frac{}{t}\left( \frac{\varphi_{s}}{\varphi_{r}} \right)} = {{{\begin{pmatrix}{{- R_{s}}L_{sg}I} & {R_{s}L_{m\quad g}I} \\{R_{r}L_{m\quad g}I} & {{{- R_{r}}L_{rg}I} + {\omega_{r}J}}\end{pmatrix} \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} + {\begin{pmatrix}I \\0\end{pmatrix} \cdot v_{s}}} = {{Ax} + {Bu}}}} \\{i_{s} = {{\left( {{L_{sg}I} - {L_{m\quad g}I}} \right) \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} = {Cx}}}\end{matrix} \right.$

where φ_(s)=[φ_(sα) φ_(sβ)]^(T): space vectors associated with a statorflux; φ_(r)=[φ_(rα) φ_(sβ)]^(T): space vectors associated with a rotorflux; i_(s)=[i_(sα) i_(sβ)]^(T): space vectors associated with a statorcurrent; i_(r)=[i_(rα) i_(rβ)]^(T): space vectors associated with astator voltage;${L_{sg} = {\frac{1}{\sigma \cdot L_{s}} = \frac{L_{r}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{rg} = {\frac{1}{\sigma \cdot L_{r}} = \frac{L_{s}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{m\quad g} = {\frac{1}{\sigma \cdot L_{m}} = \frac{L_{m}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${I = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}};\quad {J = \begin{bmatrix}0 & {- 1} \\1 & 0\end{bmatrix}};\quad {0 = \begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}};$

R_(s), R_(r): Stator and rotor resistance; L_(s), L_(r), L_(m): Stator,rotor, and mutual inductance; σ=1−L_(m) ²/(L_(s)·L_(r)): Total leakagecoefficient; ω_(r): Angular rotor speed, said observer comprising: afeedback gain determining unit determining a feedback gain K whichstabilizes an equation of estimation error of an estimated statorcurrent e_(i), $\begin{matrix}{e_{i} = \quad {{Ce} = {{C\left( {{s\quad I_{4}} - A - {K\quad C}} \right)}^{- 1}\begin{pmatrix}0 \\{- I}\end{pmatrix}\quad \left( {{- \left( {{\hat{\omega}}_{r} - \omega_{r}} \right)}J\quad {\hat{\varphi}}_{r}} \right)}}} \\{= \quad {{G_{1}(s)} \cdot \left( {{- \Delta}\quad \omega_{r}J\quad \varphi_{r}} \right)}}\end{matrix}$

 where values below the symbol {circumflex over ( )} are observedvalues; C=[L_(sg)I−L_(mg)I]; ${A = \begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}};$

A₁₁=−R_(s)L_(sg)I; A₁₂=R_(s)L_(mg)I; A₂₁=R_(r)L_(mg)I;A₂₂=−R_(r)L_(rg)I+ω_(r)J; e_(i): estimated stator current; s: Laplaceoperator; G_(l)(s): linear transfer function; and an outputting unitobtaining and outputting at least either of an observed flux and anestimated speed based on the feedback gain K.
 9. The observer accordingto claim 8, wherein said feedback gain determining unit determines thefeedback gain a magnitude of which is within a predetermined range if arotation speed of the induction motor is equal to or higher than apreset value.
 10. An observer used to vector-control an induction motorthat does not comprise at least either a speed sensor or a positionsensor, and is represented by a following equation,$\left\{ \begin{matrix}{{\frac{}{t}\left( \frac{\varphi_{s}}{\varphi_{r}} \right)} = {{{\begin{pmatrix}{{- R_{s}}L_{sg}I} & {R_{s}L_{m\quad g}I} \\{R_{r}L_{m\quad g}I} & {{{- R_{r}}L_{rg}I} + {\omega_{r}J}}\end{pmatrix} \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} + {\begin{pmatrix}I \\0\end{pmatrix} \cdot v_{s}}} = {{Ax} + {Bu}}}} \\{i_{s} = {{\left( {{L_{sg}I} - {L_{m\quad g}I}} \right) \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} = {Cx}}}\end{matrix} \right.$

where φ_(s)=[φ_(sα) φ_(sβ)]^(T): space vectors associated with a statorflux; φ_(r)=[φ_(rα) φ_(rβ)]^(T): space vectors associated with a rotorflux; i_(s)=[i_(sα) i_(sβ)]^(T): space vectors associated with a statorcurrent; i_(r)=[i_(rα) i_(rβ)]^(T): space vectors associated with astator voltage;${L_{sg} = {\frac{1}{\sigma \cdot L_{s}} = \frac{L_{r}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{rg} = {\frac{1}{\sigma \cdot L_{r}} = \frac{L_{s}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{m\quad g} = {\frac{1}{\sigma \cdot L_{m}} = \frac{L_{m}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${I = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}};\quad {J = \begin{bmatrix}0 & {- 1} \\1 & 0\end{bmatrix}};\quad {0 = \begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}};$

R_(s), R_(r): Stator and rotor resistance; L_(s), L_(r), L_(m): Stator,rotor, and mutual inductance; σ=1−L_(m) ²/(L_(s)·L_(r)): Total leakagecoefficient; ω_(r): Angular rotor speed, said observer comprising:feedback gain determining means for determining a feedback gain K whichstabilizes an equation of estimation error of an estimated statorcurrent e_(i), $\begin{matrix}{e_{i} = \quad {{Ce} = {{C\left( {{s\quad I_{4}} - A - {K\quad C}} \right)}^{- 1}\begin{pmatrix}0 \\{- I}\end{pmatrix}\quad \left( {{- \left( {{\hat{\omega}}_{r} - \omega_{r}} \right)}J\quad {\hat{\varphi}}_{r}} \right)}}} \\{= \quad {{G_{1}(s)} \cdot \left( {{- \Delta}\quad \omega_{r}J\quad \varphi_{r}} \right)}}\end{matrix}$

 where values below the symbol {circumflex over ( )} are observedvalues; C=[L_(sg)I−L_(mg)I]; ${A = \begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}};$

A₁₁=−R_(s)L_(sg)I; A₁₂=R_(s)L_(mg)I; A₂₁=R_(r)L_(mg)I;A₂₂=−R_(r)L_(rg)I+ω_(r)J; e_(i): stator current; s: Laplace operator;G_(l)(s): linear transfer function; and outputting means for obtainingand outputting at least either of an observed flux and an estimatedspeed based on the feedback gain K.
 11. A method vector-controlling aninduction motor that does not comprise at least either a speed sensor ora position sensor, and is represented by a following equation,$\left\{ \begin{matrix}{{\frac{}{t}\left( \frac{\varphi_{s}}{\varphi_{r}} \right)} = {{{\begin{pmatrix}{{- R_{s}}L_{sg}I} & {R_{s}L_{m\quad g}I} \\{R_{r}L_{m\quad g}I} & {{{- R_{r}}L_{rg}I} + {\omega_{r}J}}\end{pmatrix} \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} + {\begin{pmatrix}I \\0\end{pmatrix} \cdot v_{s}}} = {{Ax} + {Bu}}}} \\{i_{s} = {{\left( {{L_{sg}I} - {L_{m\quad g}I}} \right) \cdot \begin{pmatrix}\varphi_{s} \\\varphi_{r}\end{pmatrix}} = {Cx}}}\end{matrix} \right.$

where φ_(s)=[φ_(sα) φ_(sβ)]^(T): space vectors associated with a statorflux; φ_(r)=[φ_(rα) φ_(rβ)]^(T): space vectors associated with a rotorflux; i_(s)=[i_(sα) i_(sβ)]^(T): space vectors associated with a statorcurrent; i_(r)=[i_(rα) i_(rβ)]^(T): space vectors associated with astator voltage;${L_{sg} = {\frac{1}{\sigma \cdot L_{s}} = \frac{L_{r}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{rg} = {\frac{1}{\sigma \cdot L_{r}} = \frac{L_{s}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${L_{m\quad g} = {\frac{1}{\sigma \cdot L_{m}} = \frac{L_{m}}{{L_{s} \cdot L_{r}} - L_{m}^{2}}}};$${I = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}};\quad {J = \begin{bmatrix}0 & {- 1} \\1 & 0\end{bmatrix}};\quad {0 = \begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}};$

R_(s), R_(r): Stator and rotor resistance; L_(s), L_(r), L_(m): Stator,rotor, and mutual inductance; σ=1−L_(m) ²/(L_(s)·L_(r)): Total leakagecoefficient; ω_(r): Angular rotor speed, said method comprising thesteps of: determining a feedback gain K which stabilizes an equation ofestimation error of an estimated stator current e_(i), $\begin{matrix}{e_{i} = \quad {{Ce} = {{C\left( {{s\quad I_{4}} - A - {K\quad C}} \right)}^{- 1}\begin{pmatrix}0 \\{- I}\end{pmatrix}\quad \left( {{- \left( {{\hat{\omega}}_{r} - \omega_{r}} \right)}J\quad {\hat{\varphi}}_{r}} \right)}}} \\{= \quad {{G_{1}(s)} \cdot \left( {{- \Delta}\quad \omega_{r}J\quad \varphi_{r}} \right)}}\end{matrix}$

 where values below the symbol {circumflex over ( )} are observedvalues; C=[L_(sg)I−L_(mg)I]; ${A = \begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}};$

A₁₁=−R_(s)L_(sg)I; A₁₂=R_(s)L_(mg)I; A₂₁=R_(r)L_(mg)I;A₂₂=−R_(r)L_(rg)I+ω_(r)J; e_(i): stator current; s: Laplace operator;G_(l)(s): linear transfer function; and obtaining at least either of anobserved flux and an estimated speed based on the feedback gain K. 12.The method according to claim 11, wherein the feedback gain a magnitudeof which is within a predetermined range if a rotation speed of theinduction motor is equal to or higher than a preset value.